Abstract
UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function
Q
represented by a self-adjoint linear relation
A
in a Pontryagin space is decomposed by means of the reducing subspaces of
A
.
The sum of two functions
Q
i
∈
N
κ
i
(
ℋ
)
,
i
=
1,2
,
minimally represented by the triplets
(
𝒦
i
,
A
i
,
Γ
i
)
is also studied. For this purpose, we create a model
(
𝒦
˜
,
A
˜
,
Γ
˜
)
to represent
Q
:
=
Q
1
+
Q
2
in terms of
(
𝒦
i
,
A
i
,
Γ
i
)
. By using this model, necessary and sufficient conditions for
κ
=
κ
1
+
κ
2
are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation
A
affect the reducing subspaces of
A
and the decomposition of the corresponding function
Q
.
Publisher
Institute of Mathematics National Academy of Sciences of Ukraine
Subject
General Earth and Planetary Sciences,General Engineering,General Environmental Science
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